Fractal dimension of the controlled Julia sets of the output duopoly competing evolution model

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1 Available online at J. Math. Computer Sci. 1 (1) 1 71 Research Article Fractal dimension of the controlled Julia sets of the output duopol competing evolution model Zhaoqing Li a Yongping Zhang a Jian Liu b a School of Mathematics and Statistics Shandong Universit Weihai Weihai 9 China. b School of Mathematic and Quantitative Economics Shandong Universit of Finance and Economics Jinan 51 China. Abstract The output duopol competing evolution model has an integral role in the stud of the economic phenomenon. In this paper the basic methods of Julia sets is applied to this model. At first Julia set of this model is introduced. Then two different control methods are taken to control Julia set: one is the step hsteresis control method and the other is the optimal function control. Meanwhile bo-counting dimensions of the controlled Julia set under these methods are computed to depict the compleit of Julia sets and the sstem. The simulation results show the efficac of these methods. c 1 All rights reserved. Kewords: The output duopol competing evolution model Julia set step hsteresis control method optimal function control fractal dimension. 1 MSC: A33 E7. 1. Introduction Man comple phenomena have been eplained well b using the special properties of fractal theor. And the methods of fractal have been applied etensivel such as phsics chemistr and even economics. An analsis of stock market was presented b use of a simple fractal function and based on a detailed analsis of financial inde behavior the authors proposed a method to identif the stage of the current financial growth and estimate the time in which the inde value is going to reach the maimum []. In [11] the eistence of fractal basin boundaries has been eamined which Corresponding author addresses: qing71@13.com (Zhaoqing Li) pzhangsd@1.com (Yongping Zhang) inqingjane@13.com (Jian Liu) Received 1-3-

2 Z. Q. Li Y. P. Zhang J. Liu J. Math. Computer Sci. 1 (1) 1 71 is important to nonlinear economic sstems. In [15] the authors proposed a coordination inde of spatial distribution in the regional transportation econom b use of the fractal theor. Combined the coordination inde and other indicators the coordination degree of regional transportation econom has been reflected more comprehensivel. The spatial structure of China s vegetation has been analzed quantitativel based on fractal theor in [17]. Dimension is an important feature of geometric objects and it is the required number of independent coordinates to determine a position of geometric object. People have been accustomed to the integer dimension in Euclidean space. But in the research of clouds and some natural phenomena such as the coastline researchers found that the dimension of these cases can not use the traditional Euclidean dimension to describe. The fractal dimension is one of the most basic variables to quantitativel represent the random shapes and phenomena with self-similarit. Fractal dimension is a ver important concept in fractal theor and it is different from the Euclidean geometr dimension that shows integer degree of freedom. Dimension can be fraction in fractal theor and this kind of dimension is the important concept which is introduced b the phsicists when the stud the theor of chaotic attractor. In order to quantitativel describe the non-regular of objective things Hausdorff a mathematician studied the fractal dimension and dimension was etended from integer to fraction at the beginning of the th centur and it broke through the limit that topological sets dimension was integer. And now the measures and dimension of Hausdorff has become the core concept and mathematical foundation in the fractal theor [7]. Fractal dimension has man different definitions for different applications such as Hausdorff dimension bo-counting dimension similarit dimension and information dimension etc. In practical applications the calculation of Hausdorff dimension is ver difficult so the bo-counting dimension is often used to describe fractal. Bo-counting dimension is not onl convenient for practical applications but also has important theoretical significance and it is eas to use bo-counting dimension to describe the non-regular of fractal [ 9]. In this paper we take the bo-counting dimension to discuss. Cournot model which was put forward in 1838 b the French economist Anthon Augustine Librar was the earl oligopol model and lots of research has been promoted with this model. Based on cournot model the output duopol competing evolution model was introduced [1]. In order to get the biggest profit sellers need to make the output decisions. Under the assumption that sellers do not have entire acquaintance about the market the adjust their output just according to the local margin profit and the competitor s output decision. In [3] the different results were showed b the output duopol competing evolution model from the economic chaos control which was introduced in game theor. In this paper the basic theor and method of fractal is used into the output duopol competing evolution model to analze this model from the fractal viewpoint. This paper is mainl divided into three parts to discuss the analsis and control of the output duopol competing evolution model. Firstl the basic theor of the Julia set is given. Then appropriate parameters are selected to get the Julia set according to the model and the attractiveness of the fied point of the model is also analzed. Finall the step hsteresis control and the optimal function control are taken to control the Julia set of the model. And the bo-counting dimensions of the controlled Julia sets under these two control methods are computed respectivel to describe the compleit of the controlled Julia set of the model.. Basic theor B assuming that f is a polnomial in the comple plane f: C C f(z) = a + a 1 z a n z n n we record that f k is the k-th recombination of f that is f... f. And f k (ω) is the k-th iteration of ω that is f(f(...f(ω))). If ω satisfies f(ω) = ω we call ω as a fied point. If p is an

3 Z. Q. Li Y. P. Zhang J. Liu J. Math. Computer Sci. 1 (1) integer bigger than one and satisfies the equation f p (ω) = ω we call ω as a ccle point. And if there is a minimum p which can meet this equation we call p as the period of ω and ω f(ω)... f p 1 (ω) as a orbit of ω with period p. Assume that ω is a periodic point with period p and (f p ) (ω) = λ. We call ω to be super attractive if λ = ; attractive if λ < 1; neutral if λ = 1; repeller if λ > 1. Definition.1 ([]). Julia set J(f) is defined to be the closure of the repeller periodic points of f and the complementar set of Julia set written as F (f) is called Fatou set or stable set. In numerous fractal dimension the definition of Hausdorff based on the Caratheodor is the oldest and it ma be one of the most important definition. For an set the Hausdorff dimension can be defined but it is hard to calculate or estimate its value in practice. On the contrar bo-counting dimension has become one of the most widel used dimension because it is eas to calculate and estimate. Definition. ([]). B assuming that F is an non-empt bounded set on R n N δ (F ) is the minimum number of sets that can cover F with the maimum diameter δ. Then the upper and lower bo-counting dimension of F can be defined as follows dim B F = lim( log N δ(f ) ) log δ δ dim B F = lim δ ( log N δ(f ) log δ ). If these two values are equal this common value is called the bo-counting dimension and it can be written as dim B F = lim δ ( log N δ(f ) log δ ). 3. Julia set of the output duopol competing evolution mode Consider the output duopol competing evolution model as follows: { (t + 1) = (t) + α1 (t)[a b((t) + (t)) (b + e 1 )(t) d 1 ](1 s) (t + 1) = (t) + α (t)[a b((t) + (t)) (b + e )(t) d ](1 s) where (t) and (t) are the output of two companies at time t respectivel t = 1 a b are positive constants and a is the highest price in the market. Assume that the cost functions are C() = c 1 + d 1 + e 1 and C() = c + d + e and the 1st derivative are positive and the nd derivative are negative so we can get d i e i >. Since companies marginal profits are demanded to be less than the price of the same product in the market we have d 1 + e 1 < a d + e < a. α i are the positive parameters which show the companies adjustment speed. Julia set is a notation about the comple dnamical sstem but the model (3.1) is a real sstem and it is not suitable to consider Julia set. However if we take z(t) = (t) + i(t) the equation (3.1) can be seen as a map on the comple plane and then we build the Julia set. Let H( ) = ( + α 1 [a b( + ) (b + e 1 ) d 1 ](1 s) + α [a b( + ) (b + e ) d ](1 s)). Definition 3.1. Let K be the set of ( ) in R whose trajector is limited i.e. K = {( ) R {H n ( )} n=1is bounded}. Then the set K is called the filled Julia set corresponding the map H( ). The boundar of K is called the Julia set of the map H( ) which is denoted b J H i.e. J H = K. (3.1)

4 Z. Q. Li Y. P. Zhang J. Liu J. Math. Computer Sci. 1 (1) 1 71 For eample the parameters of model (3.1) are taken to be a = 1 b = 1 d 1 = 1 d = 1 e 1 = 1 e = 1.1 s =.3 α 1 =.5 α =.1 then the corresponding Julia set is shown in Fig. 1. Fig. 1 Original Julia sets of sstem. The curve in Fig. 1 represents the initial output of two competitors. When we take the values inside of the curve the two competitors output will be stable and bounded. Otherwise one of the competitors or both competitors output will tend to infinit that is to sa the total suppl will tend to infinit. According to the definition of the fied point we can get the fied points of sstem (3.1) as follows: (1) ( ) = ( ); () =. Then we have a b (b + e ) d = and ( ) = ( a d (b+e ) ); (3) =. Then we have a b (b + e 1 ) d 1 = and ( ) = ( a d 1 (b+e 1 ) ); (). Then we have ( ) = ( (a d 1)(b+e ) b(a d ) 3b +be 1 +be +e 1 e (a d )(b+e 1 ) b(a d 1 ) 3b +be 1 +be +e 1 e ). Here we just consider the fied point that both and are not zero. So for the eample above we can get the fied point ( ) = ( ).. Julia sets control and the change of its bo-counting dimension Chaos control has attracted attentions and man control methods have been introduced [13 1]. In recent ears control of Julia sets is discussed. In this section we will consider the control of Julia sets of model (3.1) and compute the bo-counting dimension of the controlled Julia sets to describe the compleit of Julia set and the sstem. Rewrite the model (3.1) as { (n + 1) = (n) + α1 (n)[a b((n) + (n)) (b + e 1 )(n) d 1 ](1 s) (.1) (n + 1) = (n) + α (n)[a b((n) + (n)) (b + e )(n) d ](1 s). In this section two control methods will be introduced to control Julia set of the model (.1). Let { F (n n ) = α 1 n [a b( n + n ) (b + e 1 ) n d 1 ](1 s) G( n n ) = α n [a b( n + n ) (b + e ) n d ](1 s).

5 Z. Q. Li Y. P. Zhang J. Liu J. Math. Computer Sci. 1 (1) Step hsteresis control method B introducing the control items [1] into (.1) we can get the following controlled sstem { n+1 = n + F ( n n ) + k( n n 1 ) n 3 (.) n+1 = n + G( n n ) + k( n n 1 ) n 3 where k is the control parameter. This control method starts from the third step and the second step is { = 1 + α 1 1 [a b( ) (b + e 1 ) 1 d 1 ](1 s) = 1 + α 1 [a b( ) (b + e ) 1 d ](1 s). It is clear that the fied point of the original sstem is also the fied point of the controlled sstem. To simplif the Jacobi matri of the controlled sstem the controlled sstem (.) can be rewritten as the following form n+1 = n + F ( n n ) + k( n s n ) n 3 n+1 = n + G( n n ) + k( n t n ) n 3 s n+1 = n n 3 t n+1 = n n 3. Let f( s t) = + F ( ) + k( s) g( s t) = + G( ) + k( t) p( s t) = q( s t) =. And the fied point of (.3) is ( s t ) where s = t =. Therefore the Jacobi matri of the controlled model (.3) at this fied point is J = (fgpq) (st) ( s t ) = a 1 a k a 3 a k 1 1 where a 1 = (1+k)+α 1 (1 s)a α 1 d 1 (1 s) α 1 (1 s)(b+e 1 ) α 1 b(1 s) a = α 1 b(1 s) a 3 = α b(1 s) a = (1 + k) + α (1 s)a α d (1 s) α (1 s)(b + e ) α b(1 s) and the corresponding characteristic equation is That is λe J = λ a 1 a k a 3 λ a k 1 λ 1 λ λ + cλ 3 + dλ + eλ + h = =. where c = (a 1 + a ) d = (a 1 a a a 3 ) e = k(a 1 a ) h = k. Let p 1 = (3c 8d) q 1 = 3c + 1d 1c d + 1ce h r = (c 3 cd + e) A = p 1 3q 1 B = p 1 q 1 3r C = q 1 3p 1 r. According to the discriminant the results have the following several cases. Case 1: A = B =. Let (.3) µ 1 = p 1 3 µ = q 1 p 1 and we have µ 1 = µ = µ 3. The solutions are µ 3 = 3r q 1

6 Z. Q. Li Y. P. Zhang J. Liu J. Math. Computer Sci. 1 (1) 1 71 Case : B AC =. Let where ν = B and the solutions are A Case 3: B AC <. Let where T = Ap 1 3B and the solutions are A 1.5 Case : B AC >. Let Then the solutions are: λ 1 = 1 ( c + µ 1 + µ + µ 3 ) λ = 1 ( c + µ 1 µ µ 3 ) λ 3 = 1 ( c µ 1 + µ µ 3 ) λ = 1 ( c µ 1 µ + µ 3 ). µ 1 = p 1 + ν µ = µ 3 = ν λ 1 = 1 ( c + µ 1 + µ + µ 3 ) λ = 1 ( c + µ 1 µ µ 3 ) λ 3 = 1 ( c µ 1 + µ µ 3 ) λ = 1 ( c µ 1 µ + µ 3 ). µ 1 = 1 3 (p 1 + A cos( 1 arccos T )) 3 µ = 1 3 (p 1 + A cos( 1 3 arccos T + π 3 )) µ 3 = 1 3 (p 1 + A cos( 1 3 arccos T π 3 )) 1 = d ce + 1eh λ 1 = 1 ( c + µ 1 + µ + µ 3 ) λ = 1 ( c + µ 1 µ µ 3 ) λ 3 = 1 ( c µ 1 + µ µ 3 ) λ = 1 ( c µ 1 µ + µ 3 ). = d 3 9cde + 7e + 7c h 7dh 3 1 = a λ 1 = c 1 λ = c 1 λ 3 = c + 1 λ = c + 1 c d + 1 c d 3 3 c d c d 3 3 c d + 1 c d 3 3 c d c d 3 3 c3 +cd 8e c d3 + c3 +cd 8e c d3 + c3 +cd 8e c d3 + c3 +cd 8e. c d3 +

7 Z. Q. Li Y. P. Zhang J. Liu J. Math. Computer Sci. 1 (1) Above all no matter what the solution of this quantic equation is the fied point ( s t ) is attractive as long as the value of k can satisf the inequalit: λ 1 < 1 λ < 1 λ 3 < 1 and λ < 1. Then we sa that the control of the sstem is accomplished. For eample we take the parameters in (.1) as a = 1 b = 1 d 1 = 1 d = 1 e 1 = 1 e = 1.1 s =.3 α 1 =.5 α =.1 and the corresponding original Julia set is shown in Fig. 1. The fied point of the sstem is ( s t ) = ( ) so according to the previous discussion the Jacobi matri of the control model (.3) at the fied point is J = k.35 k k k 1 1 Then the corresponding characteristic equation is λ k k λe J =.119 λ k.97 k 1 λ =. 1 λ That is λ + c λ 3 + d λ + e λ + h = where c = (k 1.15) d = k 1.15k.9 e =.151k h = k. Let p 1 = (3c 8d) q 1 = 3c + 1d 1c d + 1ce h r = (c 3 cd + e) A 1 = p 1 3q 1 B 1 = p 1 q 1 3r C 1 = q 1 3p 1 r. Let D = B 1 A 1 C 1 and with the changes of the values of the control parameter k we can get the solutions b the four cases above. The corresponding Julia sets are shown in Fig... (a) (b) (c) (d)

8 Z. Q. Li Y. P. Zhang J. Liu J. Math. Computer Sci. 1 (1) (e) (f) Fig. The changing of Julia sets of the controlled sstem: (a) k =.; (b) k =.5; (c) k =.5; k =.; (f) k =.5. (d) k =.55; (e) The changes of bo-counting dimension of the controlled Julia set with the changes of the values of the controlled parameter k are shown in Table 1 or Fig. 3. Table 1: The changes of bo-counting dimension with changes of k. k hw k hw k hw k hw k hw k hw hw k Fig. 3 The scatter diagram with the change of the control parameter and bo-counting dimension... The optimal function control B introducing the control items [] into sstem (.1) we can get the following sstem { n+1 = n + F ( n n ) + θ( n + F ( n n ) n ) n+1 = n + G( n n ) + θ( n + G( n n ) n )

9 Z. Q. Li Y. P. Zhang J. Liu J. Math. Computer Sci. 1 (1) where θ is the control parameter. Let h( ) = + F ( ) + θf ( ) l( ) = + G( ) + θg( ). The fied point does not need to be known in this control method. But in order to analze the stabilit of the fied point we also give the value of the fied point denoted b ( ). So the Jacobi matri of the control model at the fied point is J = [ h l h l ] ( ) [ c11 c = 1 c 1 c where c 11 = 1 + (1 + θ)(α 1 (1 s)a α 1 d 1 (1 s) α 1 (1 s)(b + e 1 ) α 1 b(1 s) ) c 1 = (1+θ)(α 1 b(1 s) ) c 1 = (1+θ)(α b(1 s) ) c = 1+(1+θ)(α (1 s)aα d (1 s) α (1 s)(b + e ) α b(1 s) ). The corresponding characteristic equation is λe J = λ c 11 c 1 c 1 λ c = that is ] λ (c 11 + c )λ + (c 11 c c 1 c 1 ) =. B solving this equation we can get λ 1 = c 11+c + λ = c 11+c where = (c 11 + c ) (c 11 c c 1 c 1 ). When the values of k satisf the inequalit λ 1 < 1 and λ < 1 the fied point ( ) is stable. Then we sa that the control of the sstem is accomplished. For eample we take the parameters in (.1) as a = 1 b = 1 d 1 = 1 d = 1 e 1 = 1 e = 1.1 s =.3 α 1 =.5 α =.1 and the original Julia set is shown in Fig. 1. The fied point of the sstem can be calculated from the third part ( ) = ( ). So according to the above the Jacobi matri of the control model at the fied point is [ ] 1.5(1 + θ).35(1 + θ) J =..119(1 + θ) 1.5(1 + θ) Therefore the characteristic equation is λe J = λ (1.5(1 + θ)).35(1 + θ).119(1 + θ) λ (1.5(1 + θ)) = that is λ + d 1 λ + d = where d 1 = ( 3.15(1 + θ)) d = (1 + θ) + 1.5(1 + θ). B solving this equation we have λ 1 = d 1+ and λ = d 1 where = d 1 d. When the values of θ satisf the inequalit λ 1 < 1 and λ < 1 we sa that the control of the sstem is accomplished. According to the values of the control parameter θ the corresponding Julia sets are shown in Fig.. (a) (b)

10 Z. Q. Li Y. P. Zhang J. Liu J. Math. Computer Sci. 1 (1) (c) (d) Fig. The changing of Julia sets of the controlled sstem: (a) θ =.8; (b) θ =.85; (c) θ =.9; (d) θ =.1. The changes of the bo-counting dimension of the controlled Julia set with the changes of the values of the control parameter θ are shown Table. Table : The changes of bo-counting dimension with changes of k. θ hw θ hw θ hw θ hw hw k Fig. 5 The scatter diagram with the change of the control parameter and bo-counting dimension. Though the values of θ we choose in this article are limited but from Fig. 5 the change of the bo-counting dimension with θ is heterogeneous but the change tendenc is that the bo-counting dimension gets bigger with the value of θ gets bigger. 5. Conclusion Fractal research in economics has attracted etensive attentions and applications which provides a new tool to stud some economic problems and it also has injected new vitalit to promote the

11 Z. Q. Li Y. P. Zhang J. Liu J. Math. Computer Sci. 1 (1) development of economics [ ]. As an important concept in fractal theor fractal dimension describes the compleit of things or to sa it is a measure about irregularit of comple shapes. Among numerous definitions of fractal dimension bo-counting dimension has been widel used because of its simple calculations and estimation. Based on these properties of fractal dimension it has been introduced to consider and describe the comple economic phenomenon. In this paper we use the basic method of Julia set in fractal theor to set up the initial Julia set of the output duopol competing evolution model. Then two different control methods are applied to control the original model and the corresponding bo-counting dimension of the controlled Julia set with the changes of the control parameter k under different was are also taken into consideration. The simulation results show the effectiveness of these methods. Acknowledgment This work is supported b the National Natural Science Foundation of China (Grant no and ) and China Postdoctoral Science Foundation (Grant no. 1M59188). References [1] H. N. Agiza A. S. Hegazi A. A. Elsadan The dnamics of Bowle s model with bounded rationalit Chaos Solitons Fractals 1 (1) [] L. Budinski-Petković I. Lončarević Z. M. Jakšić S. B. Vrhovac Fractal properties of financial markets Phs. A 1 (1) [3] J. G. Du T. Huang Z. Sheng Analsis of decision-making in economic chaos control Nonlinear Anal. Real World Appl. 1 (9) [] K. Falconer Fractal geometr Mathematical foundations and applications John Wile & Sons Ltd. Chichester (199) [5] Y. Lin Fractal and its application in the securities market Econ. Probl. 8 (1). 5 [] P. Liu Spatial fractal control and chaotic snchronization in comple dnamical sstem Shandong Univ. (1) 5.. [7] F. Lu Fractal theor and its application in economic management Financ. Manag. Inst. 3 (11) [8] Z. Peng C. Li H. Li The research of enterprise and fractal management model in Era of knowledge econom Business Studies 55 () [9] D. Preiss Nondifferentiable functions (Czech) Pokrok Mat. Fz. Astronom. 8 (1983) [1] J. Shu D. Tan J. Wu The eploration of fractal structure in China s stock market J. Southwest Jiaotong Univ. 38 () [11] A. S. Soliman Fractals in nonlinear economic dnamic sstems Chaos Solitons Fractals 7 (199) [1] M. de Sousa Vieira A. J. Lichtenberg Controlling chaos using nonlinear feedback with dela Phs. Rev. E 5 (199) [13] C. Wang R. Chu J. Ma Controlling a chaotic resonator b means of dnamic track control Compleit 1 (15) [1] F. Zhang C. Mu G. Zhang D. Lin Dnamics of two classes of Lorenz-tpe chaotic sstems Compleit 1 (15) [15] M. Zhang Z. Wei The Application of the fractal theor in describing the regional transportation econom coordination China soft sci. 1 (199) [1] C. Zhang F. Wu L. Zhao The application of the fractal theor in economics Statistics and Decision 1 (9) [17] X. H. Zhu N. Patel W. Zuo X. C. Yang Fractal analsis applied to spatial structure of chinas vegetation Chinese Geographical Sci. 1 ()